In The Name of God The Compassionate The Merciful

1. In The Name of God The Compassionate The Merciful

2. Wavelet Based Methodsfor System Identification Nafise Erfanian Saeedi

3. Presentation Agenda • Introduction to wavelets • General applications for wavelets • Application of wavelets in system identification • Simulation Example • Comparison with conventional methods • Conclusions

4. Introduction to wavelets A wavelet is a waveform of effectively limited duration that has an average value of zero

5. Introduction to wavelets • Wavelet Analysis • Comparing wavelet analysis to Fourier analysis

6. Introduction to wavelets Introduction to wavelets Continues Wavelet Transform (CWT) Wavelet Transform Discrete Wavelet Transform (DWT) `

7. Introduction to wavelets Introduction to wavelets Continues Wavelet Transform

8. Introduction to wavelets Introduction to wavelets Five Steps to CWT 1- Take a wavelet and compare it to a section at the start of the original signal. 2- Calculate a number, C, that represents how closely correlated the wavelet is with this section of the signal.Note that the results will depend on the shape of the wavelet you choose.

9. Introduction to wavelets Introduction to wavelets 3- Shift the wavelet to the right and repeat steps 1 and 2 until you've covered the whole signal.

10. Introduction to wavelets Introduction to wavelets 4- Scale (stretch) the wavelet and repeat steps 1 through 3. 5- Repeat steps 1 through 4 for all scales.

11. Introduction to wavelets Introduction to wavelets Results Large Coefficients Scale Small Coefficients Time

12. Introduction to wavelets Introduction to wavelets Low scale >> Compressed wavelet >> Rapidly changing details >> High frequency High scale >> Stretched wavelet >> Slowly changing, coarse features >> Low frequency

13. Introduction to wavelets Introduction to wavelets An Example from Nature: Lunar Surface

14. Introduction to wavelets Introduction to wavelets Discrete Wavelet Transform Approximations and Details One Stage Filtering Problem: Increasing data volume

15. Introduction to wavelets Introduction to wavelets Filtering with down sampling

16. Introduction to wavelets Introduction to wavelets Multi Stage Decomposition

17. Morlet Haar Mexican hat PDF’s Derivative Mayer Symlet Coiflet Daubechies Introduction to wavelets Introduction to wavelets Different Mother wavelets

18. Introduction to wavelets General Applications for wavelets 1) Detecting Discontinuities and Breakdown Points Freqbrk.mat db5 level 5

19. Introduction to wavelets General Applications for wavelets 2) Detecting Long-Term Evolution Cnoislop.mat db3 level 6

20. Introduction to wavelets General Applications for wavelets 3) Detecting Self-Similarity vonkoch.mat coif3 continues

21. Introduction to wavelets General Applications for wavelets 4) Identifying Pure Frequencies sumsin.mat db3 level 5 2 Hz 20 Hz 200 Hz

22. Introduction to wavelets General Applications for wavelets 5) De-Noising Signals noisdopp.mat sym4 level 5 Problem: Loss of Data

23. Introduction to wavelets General Applications for wavelets Solution: Special Algorithms

24. Introduction to wavelets General Applications for wavelets Other Applications: • Biology for cell membrane recognition, to distinguish the normal from the pathological membranes • Metallurgy for the characterization of rough surfaces • Finance (which is more surprising), for detecting the properties of quick variation of values • Detection of short pathological events as epileptic crises or normal ones as evoked potentials in EEG (medicine) • Study of short-time phenomena as transient processes • Automatic target recognition

25. Introduction to wavelets Wavelets in system identification Here, we consider wavelet approaches to analyze signals that are a (linearly) filtered version of some source signal with the purpose of identifying the characteristics of the filtering system.

26. Introduction to wavelets Wavelets in system identification System Identification Methods: Parametric Non parametric

27. Introduction to wavelets Wavelets in system identification Solution one: For a causal system Problem: Round-off errors accumulate with larger time indices, making this approach impractical for slowly decaying (i.e., infinite) impulse response functions.

28. Introduction to wavelets Wavelets in system identification Solution two: Frequency-domain methods for linear systems based on coherence Analysis Usually with pseudorandom noise as input

29. Introduction to wavelets Wavelets in system identification Wavelet representation of signals For a finite energy signal: discrete parameter wavelet transform (DPWT) analyzing functions scale index k translation index m

30. Introduction to wavelets Wavelets in system identification Dyadic Sampling: compression/dilation in the DPWT is by a power of two with

31. Introduction to wavelets Wavelets in system identification DPWTs are calculated from Analysis equation For orthogonal wavelets An interesting observation

32. Introduction to wavelets Wavelets in system identification For a source-filter model

33. Introduction to wavelets Wavelets in system identification Using orthogonality property

34. Introduction to wavelets Wavelets in system identification It is proved that k=0 is the best choice to prevent aliasing without wasting resources

35. Introduction to wavelets Wavelets in system identification Discrete time signals Discrete Wavelet Transform (DWT)

36. Introduction to wavelets Wavelets in system identification System identification using DWT y[n]=h[n]*x[n] hestimated[n] x[n] excitation System under test D W T

37. Introduction to wavelets Simulation Example i) Choice of excitation System under test: Chebyshev,IIR,10th order high pass filter with 20db ripple Excitations:

38. Introduction to wavelets Simulation Example Results for different excitations Haar and Daubechies excitations give very good identification

39. Introduction to wavelets Simulation Example Results of changing the coefficients number for Daubeshies

40. Introduction to wavelets Simulation Example ii) Different Systems wavelet used as excitation and analysing function: Daubechies D4

41. Introduction to wavelets Simulation Example System 1: FIR band-stop filter (a) Frequency response (b) Error variation with frequency

42. Introduction to wavelets Simulation Example System 2: Butterworth IIR, 10th order Band-stop (a) Frequency response (b) Error variation with frequency

43. Introduction to wavelets Simulation Example System 3: Chebyshev IIR, 10th order Band-stop (a) Frequency response (b) Error variation with frequency

44. Introduction to wavelets Simulation Example System 4: Elliptic IIR, 10th order Band-stop (a) Frequency response (b) Error variation with frequency

45. Introduction to wavelets Comparison with conventional methods • Chirp method System under test: Chebyshev high-pass filter

46. Introduction to wavelets Comparison with conventional methods 2) Time domain recursion System under test: Chebyshev high-pass filter

47. Introduction to wavelets Comparison with conventional methods 3) Inverse filtering System under test: Chebyshev high-pass filter

48. Introduction to wavelets Comparison with conventional methods 4) Coherence System under test: Chebyshev high-pass filter

49. Introduction to wavelets Conclusions • A new method for non-parametric linear time-invariant system identification based on the discrete wavelet transform (DWT) is developed. • Identification is achieved using a test excitation to the system under test, that also acts as the analyzing function for the DWT of the system’s output. • The new wavelet-based method proved to be considerably better than the conventional methods in all cases.

50. Introduction to wavelets Refrence 1- R.W.-P. Luk a, R.I. Damper b, “Non-parametric linear time-invariant system identification by discrete wavelet transforms”, Elsevier Inc,2005 2- M. Misiti, Y. Misiti, G. Oppenheim, J. M. Poggi, “Wavelet Toolbox for use with matlab” Mathworks Inc., 1996. 3- کاشانی، حامد، ” کاربرد موجک در شناسايي سيستم“؛ سمينار درس مدلسازی،1383